The problem of network-constrained averaging is to compute the average of a collection of a set of values distributed throughout a network using an algorithm that can pass messages only along edges of the network. We study this problem in the noisy setting, in which the communication along each link is modeled by an additive white Gaussian noise channel. We propose a two-phase decentralized stochastic algorithm, and we use stochastic approximation methods to analyze how the number of iterations required to achieve mean-squared error d scales as the number of nodes n in the graph. Previous results provided guarantees with the number of iterations scaling inversely with the spectral gap of the graph (second smallest eigenvalue of the Laplacian). In this paper, we prove that our proposed algorithm reduces this graph dependence, up to logarithmic conditions, to the graph diameter, which cannot be improved upon by any algorithm.